9.E: Spin Angular Momentum (Exercises)
 Page ID
 15783

Find the Pauli representations of \(S_x\), \(S_y\), and \(S_z\) for a spin1 particle.

Find the Pauli representations of the normalized eigenstates of \(S_x\) and \(S_y\) for a spin\(1/2\) particle.

Suppose that a spin\(1/2\) particle has a spin vector that lies in the \(x\)\(z\) plane, making an angle \(\theta\) with the \(z\)axis. Demonstrate that a measurement of \(S_z\) yields \(\hbar/2\) with probability \(\cos^2(\theta/2)\), and \(\hbar/2\) with probability \(\sin^2(\theta/2)\).

An electron is in the spinstate \[\chi = A\,\left(\begin{array}{c}12\,{\rm i}\\2\end{array}\right)\] in the Pauli representation. Determine the constant \(A\) by normalizing \(\chi\). If a measurement of \(S_z\) is made, what values will be obtained, and with what probabilities? What is the expectation value of \(S_z\)? Repeat the previous calculations for \(S_x\) and \(S_y\).

Consider a spin\(1/2\) system represented by the normalized spinor \[\chi =\left(\begin{array}{c}\cos\alpha\\\sin\alpha\,\exp(\,{\rm i}\,\beta)\end{array}\right)\] in the Pauli representation, where \(\alpha\) and \(\beta\) are real. What is the probability that a measurement of \(S_y\) yields \(\hbar/2\)?

An electron is at rest in an oscillating magnetic field \[{\bf B} = B_0\,\cos(\omega\,t)\,{\bf e}_z,\] where \(B_0\) and \(\omega\) are real positive constants.

Find the Hamiltonian of the system.

If the electron starts in the spinup state with respect to the \(x\)axis, determine the spinor \(\chi(t)\) which represents the state of the system in the Pauli representation at all subsequent times.

Find the probability that a measurement of \(S_x\) yields the result \(\hbar/2\) as a function of time.

What is the minimum value of \(B_0\) required to force a complete flip in \(S_x\)?

Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
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